Path Planning for Multiple Robots

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Path Planning for Multiple Robots
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Basic Overview

• This paper presents a coordinated approach to
  multiple robot path planning as opposed to the
  usual decoupled planning
• consider all of the simple robots together to
  be one composite robot and compute the
  coordinated path for the composite robot
• simple roadmap constructed for one robot
• n of these roadmaps combined into a roadmap
  for the composite robot consisting of n simple
  robots

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• composite roadmap is used for retrieval of
  coordinated paths
• two composite roadmap structures
• flat super-graph
• multi-level super-graph

• proper construction of simple roadmaps
  guarantees that the resulting multi-robot planner
  can be solved in a finite amount of time
• paper applies the composite roadmap approach
  to car-like robots

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General Approaches

• Centralized
• - treat separate robots as one composite system
  - configuration space is
  formed by combining the configuration spaces of
  the individual robots
• Advantage allow for complete planners (always
  able to find a solution if one exists)
• Disadvantage much to computationally expensive
  to handle more complex problems

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General Approaches
2. Decoupled - first generate paths for the
separate robots independently and then considers
the interactions between the robots Advantage
much more acceptable time complexity than
the centralized approach Disadvantage it is
not complete and will often lead to
deadlock situations when handling more
complex problems
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General Approaches
3. Weaker Centralization Roadmap
Coordination - building and searching a data
structure that represents the Cartesian product
of the separate roadmaps. -This approach can
solve complicated problems that cannot be solved
by decoupled planners and would consume too much
time and memory using previously existing
complete planners. -Completeness while yielding
planners with acceptable time complexity
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G-Discretized Coordinated path
Concatenation of a finite number of trivial
coordinated paths
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Flat Super-Graph

• Each node corresponds to a feasible placement of
  the n simple robots at the nodes of the
  G-discretized path.
• Each edge corresponds to a trivial coordinated
  path
• Any path in the flat super-graph describes a
  G-discretized coordinated path.
• Graph search of the flat super-graph can be done
  to find G-discretized coordinated paths for the
  composite robot.

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Flat Super-Graph
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Coordinated Retractions

• Coordinated path moving each simple robot from
  their start or goal point to a node of the
  underlying simple roadmap
• Compute a coordinated retraction (Ps1,Psn) for
  the start configuration (s1,sn)
• Compute a coordinated retraction (Pg1,Pgn) for
  the goal configuration (g1,gn)
• Find the shortest path, PF, between node
  (Ps1,,Psn) and node (Pg1,,Pgn)
• The path, P is the concatenation of (Ps1,,Psn),
  PF, and P(g1,,Pgn)

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Multi-Level Super-Graph

• Reduce the size of the super-graph by combining
  multiple node-tuples into single super-nodes
• Leaves out nodes corresponding to placements
  where the robots are far apart and in no danger
  of collision
• At each step, identify the maximum subgraphs such
  that the robots are not near each other.
• Only have to consider the specific positions of
  the robots if two robots, while staying in their
  respective subgraphs, would be in danger of
  blocking each others motions.

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Multi-Level Super-Graph

• Problem of moving two robots (A1,A2) from
  configuration (x7,x3) to (x2,x4) is solved by the
  following path
• (x7,x3),(x4,x3),(x4,x1),(x2,x1),(x2,x3),(x2,x4)
• With multi-level super-graph there are two
  connected subgraphs A and B (induced by
  x1,x2,x3 and x4,x5,x6,x7 respectively. Now
  the motion can be described by the following
  reduced path
• (B,A),(x2,x1),(x2,x3),(A,B)

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Multi-Level Super-Graph
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Multi-Level Super-Graph
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Multi-Level Super-Graph
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Path Retrieval Algorithm
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Path Smoothing

• Cut out redundant path segments
• Combine alternating simple robot motions into
  simultaneous ones.

• No absolute stop criterion
• typical criteria are to smooth for a fixed
  amount of time or up to the point where no
  significant gain is any longer achieved

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Simulation Results when applied to car-like robots
Only results for the multi-level super-graph
planner were presented since the flat super-graph
method did not yield practical results for more
than three robots.
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Simulation Results when applied to car-like robots
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Simulation Results when applied to car-like robots

• Super-graph size increases with the number of
  robots
• Size of the simple roadmap does not have great
  impact on the size of the multi-level super
  graph.
• For five robots the super-graph size decreased
  with increasing size of the simple roadmaps due
  to the sieving algorithm, which prunes the
  super-graph structure.

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Simulation Results when applied to car-like robots
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Simulation Results when applied to car-like robots
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Simulation Results when applied to car-like robots